CORCORAN-HAZEN-MASOUD: HAZEN’S EXISTENTIAL-IMPORT LEMMA

Existential-import mathematics: negated-consequent qualifications.  We build on Corcoran-Masoud 2014 [1] and 2015 [2]. Several questions were left open including the question treated here:Which universalized conditionals with existential import are logically equivalent to universalized conditionals without existential import?See [1], p.53 and [2], p.10. We had previously proved that every universalized conditional without existential import is logically equivalent to some universalized conditional with existential import ([1], p.54, [2], p.10).  The universalized conditional ∀x [(S(x) &∼ P(x)) → P(x)] is the negated-consequent qualification [NCQ] of the universalized conditional ∀x [S(x) → P(x)]. It is easy to see that the NCQ of a universalized conditional is logically equivalent to the universalized conditional itself.  Moreover, the existentialized conjunction corresponding to an NCQ, say, ∃x [(S(x) &∼ P(x)) & P(x)], is evidently inconsistent and thus not implied by its NCQ unless the latter is inconsistent. This shows that no consistent negated-consequent qualification has existential import.  But every universalized-conditional is logically equivalent to an NCQ. This proves the Hazen Lemma: every consistent universalized conditional is logically equivalent to a universalized conditional without existential import.  One important question mentioned above—answered definitively by Allen Hazen in correspondence with Corcoran and Masoud—concerns which universalized conditionals with existential import are logically equivalent to universalized conditionals without existential import.   Hazen’s Theorem: A universalized conditional with existential import is logically equivalent to some universalized conditional without existential import iff it is consistent.  Hazen’s contributions nicely complement our results. They use none of our previously published conclusions: they are entirely new.  References [3] and [4] review [1] and [2], respectively.  [1] JOHN CORCORAN AND HASSAN MASOUD, Existential-import today, History and Philosophy of Logic, vol.  36 (2014), pp. 39–61.  [2] JOHN CORCORAN AND HASSAN MASOUD, Existential-import mathematics, Bulletin of Symbolic Logic, vol. 21 (2015), pp. 1–14.  [3] JONAS ARENHART, Review of Corcoran-Masoud 2014, Mathematical Reviews, 3323750, 2016.  [4] RODRIGO FREIRE, Review of Corcoran-Masoud 2015, Mathematical Reviews, 3333458, 2016.